Charlottesville, Virginia
Journal of Coastal Research
The Bruun Rule of Erosion by Sea-Level Rise: A Discussion
on Large-Scale Two- and Three-Dimensional Usages
Per Bruun
34 Baynard Cove Road
Hilton Head Island SC 29928
ABSTRACT
~,"II"
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.---=_c:s==-+ ;-••
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BRUUN, P. 1988. The Bruun Rule of Erosion by sea-level rise: A discussion on large-scale two-
and three-dimensional usages. Journal of Coastal Research, 4(4), 627-648. Charlottesville (Virginia). ISSN 0749-0208 .
This a~ticle r~views all ?asic assum.ptions for proper use of the Bruun-Rule of erosion by sealevel rise. It disproves misuses and discusses expansions of the rule's applicability in large-scale
two and three dimensions.
ADDITIONAL KEY WORDS: Bruun Rule, sea level rise, shore erosion, bottom profile development.
INTRODUCTION
The Bruun Rule of erosion, so named
by American coastal geomorphologists
(SCHWARTZ, 1967), was first published in
1962 (BRUUN, 1962). Concerning a long-term
budget of onshore/offshore movement of material, the rule is based on the assumption of a
closed material balance system between the (1)
beach and nearshore and (2) the offshore bottom
profile. Figure 1 is a schematic of the effect, a
translation of the beach profile by a distance S
following a rise a of the sea level, resulting in
a shore erosion and a deposition of sediments.
This topic is dealt with extensively in theory
(HALLERMEIER, 1972; ALLISON, 1980;
BRUUN, 1980, 1983) and through observations
in the field (BRUUN, 1956 a, b, 1962, 1980,
1983; DUBOIS, 1976; ROSEN, 1978, 1980;
WEGGEL, 1979; FISHER, 1980; HANDS, 1980;
SCHWARTZ, 1965, 1967, 1979). Most lately the
Rule has been used for various reports, e.g. the
report on the erosion at Ocean City, Maryland,
"Potential Impact of Sea Level Rise on the
Beach at Ocean City, MD", published by the
EPA, October 1985 and in a paper by EVERTS
(1985).
The "rule" has sometimes been used rather
indiscriminately without realizing its limitations. One should always remember that it is
basically two-dimensional, but it is (almost)
88001 received 5 January 1988; accepted in revision 13 May 1988.
always applied three-dimensionally. This has
caused a number of misinterpretations. Used
objectively and correctly, the rule, however,
offers several possibilities for better understanding of three-dimensional processes and for
the explanation of three-dimensional largescale coastal developments using the rule as a
kind of "base-line" for whatever development
that takes place (plus or minus) in relation to a
basic profile of well defined geometry and to the
observed relative sea level.
This paper discusses boundary conditions,
deviations and adjustments which make the
rule useful for interpretation of the observed
phenomena in quantifiable terms.
REVIEW OF BASIC ASPECTS OF
PROFILE DEVELOPMENT IN RELATION
TO SEA-LEVEL RISE
Consider a sea-level rise a in a theoretical
profile with geometrical characteristics as
shown in Figure 1. In a coastal geomorphological sense this corresponds to a lift of the profile
of the value a. For a profile y = ftx), in order to
re-establish the old profile one needs a deposition of:
~
=
f
[flx) + a] dx -
l'
f(x) dx
=
I· a
(1)
To establish quantitative equilibrium it is
assumed that:
(1) Full profile equilibrium exists, which
Bruun
628
means the combined beach and offshore profile
maintains an equilibrium shape, although seasonal fluctuations may occur. "Seasonal" means
"short term" but eq. 1 refers to the long-term
development. The outermost part of the profile
Figure 1, is a "ramp".
(2) The shore or section of shore, to which the
Rule is applied is in a quantitative materialsbalance condition, or integrated over the profile
equals zero, when M = m, + m 2 ••• + m n is the
total quantity of material moving in or out of
the profile in all directions under the action of
waves, currents and winds. To this man's activities in dredging could be added (ALLISON,
1981).
As the profile system is assumed to be in an
overall cross-sectional equilibrium, the only
way in which the material for deposition can be
obtained is by a shoreward movement. In a
practical close approximation such movement
may be determined from the equation:
~
=
l'
flxldx + h . s -
l'
f(x)dx
=
h .s
(2)
when h is "the maximum depth of exchange of
material between the nearshore and the offshore", while 1 is the length of the profile of
exchange. From eqs. 1 and 2 one has:
s=l'a/h
(3)
As pointed out by BRUUN (1962), neither the
slope of the profile, nor the point of intersection
of the new and the old profile, nor the position
of, or the seaward slope angle of the offshore bar
were used or needed for the derivation of the
above simple formula. This is the "advantage"
of the theory. As stated by ALLISON (1980), the
exactness of the theory, however, depends upon
the sll ratio (Figure 1), but this ratio is always
very small. Allison states:
"For small ratio sll < 1 there is, consequently, no need to know any detail of bottom profile at all and Bruun's Formula (1)
correctly reflects this, not containing any
parameters describing the shape of the bottom profile. Hence, the value hll in formula
(3) (having nothing to do with the slope of
the bottom profile) is, to a zero order
approximation, an invariant, valid for any
profile shape for calculation of the ratio al
s. It is proposed, therefore, that this value
h/l be known as "Bruun's Invariant."
The above, however, should not be interpreted
too rigorously. It refers to simple profile geometries (BRUUN, 1954 a,b, 1962, 1980, 1983)
and restrictions on materials, as mentioned
later must also be accepted.
RELATIVE MOVEMENTS OF PROFILES,
INCLUDING SEA-LEVEL RISE AND
FALL, TECTONIC, AND GLACIAL
MOVEMENTS
Apparent sea-level rise varies in different
parts of the world's oceans, influenced as they
are by local trends of temperature, winds, and
currents. Along the US eastern seaboard rises
averaged about 3 millimeters per year during
the last decades. Table 1 gives an impression of
these movements. A very comprehensive work
by LISLE (1982, sponsored by the Office of
Naval Research) is mentioned in the latter part
of this paper under "Latest Development in
Research."
Deviations from the Simple Rule: How to
Interpret and Quantify Them Properly
In its simplest form the rule refers to a shore
of infinite length and of neutrality of longshore
movement of material. Consequently the beach
and offshore bottom profiles maintain their geometrical shape which is solely a function of
wave action, tides and sea level movements and
materials. If wave action is always perpendicular to the shoreline there will be no resultant
or predominant longshore drift. If the water
table (apart from regular tidal action) stays
constant, the profile develops an equilibrium
shape with steepness corresponding to bottom
material characteristics and wave action, as
discussed in the following section.
Attempts have been made to compute the geometrical shape of such profiles based on idealized assumptions. There are basically two different approaches: one is of semi-theoretical
"philosophical" nature using simplified, but
still rational basic assumptions, and the other
is a detailed hydrodynamic approach that considers the equilibrium condition for a single
grain on the bottom, ignoring bottom configurations like ripple marks. The following discussion briefly explains both methods.
BRUUN's approach (1954, b, c, and 1985 with
Journal of Coastal Research, Vol. 4, No.4, 1988
The Bruun Rule of Erosion
629
y
~
a
---------------------SEA LEVEL RISE
T
h
d
~--------a DEPOSITION I
To
x
Rarop
1+------------
I
Figure 1. The Bruun Rule-translation of the beach and bottom profile resulting in shore recession and deposition of sediments
(Bruun, 1983).
Table 1. Average sea-level rises, 1940-1970 on the US east
coast (Bruun, 1973, after compilation by Hicks)
Location
Eastport, Maine
Portsmouth, New Hampshire
Woods Hole, Massachusetts
Newport, Rhode Island
New London, Connecticut
New York, New York
Sandy Hook, New Jersey
Baltimore, Maryland
Washington, D.C.
Portsmouth, Virginia
Charleston, South Carolina
Fort Pulaski, Georgia
Mayport, Florida
Miami Beach, Florida
Pensacola, Florida
Eugene I., Louisiana
Gal veston, Texas
Rate (cm/yr)
1930-1969
1927-1970
1933-1970
1931-1970
1939-1970
1893-1970
1933-1970
1903-1970
1932-1970
1936-1970
1922-1970
1936-1970
1929-1970
1932-1970
1924-1970
1940-1970
1909-1970
0.338
0.165
0.268
0.210
0.229
0.287
0.457
0.259
0.244
0.341
0.180
0.198
0.155
0.192
0.040
0.905
0.430
The trend after 1970 has generally been up but with
no definite sign of acceleration (Pirazzcli, 1986).
SCHWARTZ), an example of the former, is as
follows:
(a) The profile is formed by shear stress due
to wave action and is at right angles to the
shoreline. The material detached by the oscillating water is removed by longshore currents.
As the shear stress due to wave action in gen-
eral-and particularly during storms-is far
greater than the shear stress originating from
the longshore currents, this assumption seems
logical.
(b) In the equilibrium profile the shear stress
per unit bottom area may be assumed to be constant, i.e. the "condition" at the bottom is the
same (dr/dx = dr/dt = 0). Confirmation of this
assumption only can be attained by experiments. One obtains T = Kpu 2 ave' where P is the
density, K the resistance coefficient and u the
water velocity. If T is assumed a constant, then
fLave ~ H'lT/T sinh 2'lTy/L is also constant where T
is the wave period; H, the wave height; L, the
wave length, and y, the water depth.
(c) dE,/dx = constant, where E, is the transported wave energy per unit area of the wave,
and x is the distance from the shoreline. The
loss of energy is mainly by bottom friction, a
loss by spilling of the wave and a loss by internal friction are very small. The correctness of
this assumption can only be proven by experiments. Calcuations give:
X
Y
Y)
=L'vm=
z-rry {2(2'lT
L + !(2'lT
3 L
+ ~('lTY)
180 L ... }
)2
0
o
0
0
where y is the water depth and L, the deep
water wave length. The series is convergent for
Journal of Coastal Research, Vol. 4, No.4, 1988
Bruun
630
y < Lo/8, i.e., for storm waves on the Danish
west coast out to depths of about 12 m (40 feet)
where La = 100 m (300 feet). Since y«L a the
equation may be reduced to
y3/2
=
px
(4)
where p is a "constant" related to exposure and
bottom materials. If it now is assumed that the
loss of energy is due only to bottom friction and
that this loss per unit area e, is constant, then
T
(5)
KptJ.;ve
where K is a constant times (a/R)3/4, a is the
length of the ripple marks and R is the half
amplitude of the oscillating water motion at the
bottom (R» a, BAGNOLD, 1946). Calculations similar to those described above then give:
3/2
Y
=
P,' X
T 2 /3
(y < about Lj8)
(6)
This profile is similar to the one above (eq. 4).
Certainly the profile depends on the wave
period T, but as the profile is shaped mainly by
storm waves and as the variation in T for these
is small, the profile in reality will be the same
as that given by (4). BRUUN (1954b, 1954c)
found confirmation of this profile geometry on
the Danish North Sea Coast as well as in southern California. P, is calibrated to local environmental conditions (waves, materials).
For the area inside the breaker zone BRUUN
(1987 -88) developed the equation
yO/4
(7a)
P2 . X
That was based on model research by VELLINGA (1985). Combining the equal shear
stress requirement with BAGNOLD's above
mentioned expression for friction and using
LOSADA and DESIRE's (1985) results, by
which the amplitude is replaced by a linear
function for grain diameter, D, DRUUN (19861987) found the relation:
d O/ l 2 • D 3/4 = constant (d = y = water depth)
(7b)
Equation (7b) has to some extent been confirmed by field results in Denmark, Iceland and
Australia. D is taken as DoD. Variances are
explained
An example of a detailed hydrodynamic
approach is given by EAGLESON, GLENNE
and DRACUP (1961). The result of their
approach and computations is shown in Figure
2 and may be summarized as follows, referring
to a single grain of well defined size, geometry,
and specific gravity located on a straight slope
on other grains as indicated in Figure 2 and
subjected to a specific wave action:
(1) There is a point of "incipient motion"
when the forces are just able to initiate movement.
(2) The motion may be either up- or downslope. Wave motion in the nearshore zone is
always assymetrical with a tendency to shoreward predominance (as demonstrated by field as
well as laboratory experiments.) At one point,
theoretically speaking, an equilibrium condition between forces working upslope and forces
working downslope exists. The direction of
movement of any grain depends upon the relative location of the point of incipient motion and
the point of equilibrium condition (the null
point). If, as shown in Figure 2 the point of
incipient motion is located at a greater depth in
the profile than the point of oscillating equilibrium, material will migrate in an onshore
direction from points inside the point of oscillating equilibrium but offshore outside the said
point. This means that the profile, as a whole,
flattens (winter profile). If the opposite is the
case and the point of incipient motion is located
at less depth in the profile than the point of
oscillating equilibrium, all motion inside the
point of incipient motion will be toward the
shore, which means that the profile steepens
(summer profile). See also BRUUN (1954b,
1954c), INMAN & RUSNAK (1956), and
SWART (1974). This theory has practical
aspects and confirms the "theory" by CORNAGLIA (1887) that there is a "null point" for each
grain size on the nearshore bottom. As pointed
out by MURRAY (1966) Eagleson's theory
mainly refers to bed transport. From his field
studies in Buzzard's Bay, Massachusetts, he
concluded that "within the experimental range
of the data it is concluded that under the same
wave conditions, finer grain sizes have a
greater tendency to move offshore than coarser
grains. A change in wave state resulting in an
increase in the maximum horizontal velocity
near the bottom produces an increase in the
tendency for all test grains to move seaward".
Journal of Coastal Research, Vol. 4, No.4, 1988
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631
The Bruun Rule of Erosion
O-CONST.
FORCE,F
OR
ANGLE,.p
GRAVITY FORCE F
SIN«
O~~~=
1/2
Figure 2.
-BREAKER
o
Stability situation for a single grain on a uniform slope (from Eagleson, Glenne and Dracup, 1961).
Discussion of the Geometric Shape of the
Bottom Profile
The relationship y = px'", where m = 2/3 as
proposed in BRUUN's theory, was investigated
by DEAN (1977) who, based on a study of more
than 500 beach profiles (BRUUN had about 30),
found an equilibrium profile similar to
BRUUN's with m = 2/3. DEAN showed that
this profile, based on linear wave theory, was
consistent with uniform wave energy dissipation per unit volume due to wave breaking.
This, however, is a rather unrealistic assumption outside the surf zone.
Comparing a storm situation with a (low)
swell condition: during the storm the point of
incipient motion will be located far offshore
while the point of equilibrium will be found
closer to shore. Consequently, material will
move offshore. Conversely, during a (low) swell
situation the point of incipient motion will be
fairly close to shore while the point of oscillating equilibrium will be located further offshore.
Consequently material will move onshore from
a certain depth.
THE DEVELOPMENT OF THE
BRUUN RULE
For some years, Bruun had been concerned
with equilibrium beach profiles on the coasts of
Denmark (BRUUN, 1954a), southern California (1954b), and Florida (BRUUN 1955). He
offered the definition, "An equilibrium beach
profile is a statistical average profile which
maintains its form apart from small fluctuations incl uding seasonal fluctuations". This
concept was used in a later analysis of sea level
rise as a cause of shore erosion (BRUUN, 1962),
when Bruun hypothesized (Figure 1) that,
given an equilibrium beach profile, a rise in sea
level would be followed by: (a) a shoreward displacement of the beach profile as the upper
beach is eroded; (b) movement of the material
eroded from the upper beach would be equal in
volume to the material deposited on the near
offshore bottom; and (c) a rise of the near offshore bottom as a result of this deposition,
equal to the rise in sea level, thus maintaining
a constant water depth in that area. This proposition was essentially intuitive, although
equilibrium forms were tested by field surveys
(BRUUN 1954a), SCHWARTZ (1965) undertook laboratory wave-basin experiments to test
the validity of the hypothesis. Utilizing different wave parameters and varying amounts of
sea level change, measurements were made
before and after each run to determine the
water depth in the nearshore zone, and thus
document profile translation and erosion-deposition relationships. These elementary experiments showed support for Bruun's hypothesis.
A field study of the effects of sea level rise,
based on investigations of the response to the
effective rise in sea level occurring between
high neap and high spring tides, was conducted
on two Cape Cod beaches in the summer of 1964
(SCHWARTZ, 1979). The two beaches were the
Journal of Coastal Research, Vol. 4, No.4, 1988
Bruun
632
Nauset Light Beach and the Herring Cove
Beach, within the Cape Code National Seashore
park, providing, respectively, an open ocean
and a protected bay regime. Starting points for
profile measurements were the protected-beach
signs at each beach. The profiles were surveyed
throughout the summer using a modified version of EMERY's two-stick-profile method
(EMERY, 1961), in conjunction with SCUBA
gear and enough weights to maintain negative
buoyancy. Variations in profiles for a series of
high neap to high spring events supported the
hypothesis under investigation and it was proposed "that the concept henceforth be known as
Bruun's Rule" (SCHWARTZ, 1967). The term
Bruun Rule, first appeared in the coastal literature in an article by SWIFT (1968). This was
followed closely in books by BIRD (1969),
Coasts, and KING (1972) Beaches and Coasts.
In 1972, FISHER included the Nauset Light
and Herring Cove beach sites, together with a
discussion of the early Bruun Rule research, in
his guide to the geology of the Cape Cod
National Seashore. The rule found its way into
the Soviet literature in 1973 via KAPLIN's
(1973) Recent History of the Coasts of the World
Oceans. Further testing and refinement of the
rule followed in DUBOIS (1975, 1976, 1977),
HANDS (1976,1977,1979), and ROSEN (1978).
The history of the hypothesis and research has
been summarized by SCHWARTZ & MILICIC
(1978, 1980a, 1980b).
In November of 1979 the International Geographical Union's Commission on the Coastal
Environment held a Bruun-Rule Symposium in
Newport, Rhode Island (SCHWARTZ & FISHSER, 1980). Susbsequent literature dealing
with the Bruun Rule included ALLISON, CARBON & LICHTFIELD (1982), ALLISON &
SCHWARTZ (1981a, 1981b), BRUUN (1983),
HANDS (1983) and LEATHERMAN (1983).
Furthermore, in connection with a recent Environmental Protection Agency study of the
effects of sea level rise, the Bruun Rule and
many of the aforementioned publications have
been discussed by KANA, MICHEL, HAYES &
JENSEN (1984), LEATHERMAN (1984),
LEATHERMAN, KEARNY & CLOW (1983),
and TITUS & BARTH (1984).
Beach Erosion, Why and How
Beach erosion is the result of anyone or more
of the following adverse conditions (BIRD 1983;
BRUUN 1973): (1) The effects of human impact,
such as construction of artificial structures,
mining of beach sand, offshore dredging, or
building of dams on rivers; (2) losses of sediment offshore, onshore, alongshore and by
attrition; (3) reduction in sediment supply due
to decelerating cliff erosion; (4) reduction in
sediment supply from the sea floor; (5)
increased storminess in coastal areas or
changes in angle of wave approach; (6) increase
in beach saturation due to a higher water table
or increased precipitation; and (7) sea level rise.
These conditions are subject to large variations as they are highly dependent on many
external factors. The one aspect that will be
dealt with here, as a cause of beach erosion, is
"sea level rise".
Material Budgets
To evaluate beach erosion quantitatively
requires the establishment of a materials
budget; which means a total account of all
movement of material within an area limited
up and down the profile by boundary lines
where erosion or accretion is approximately
zero, and on the sides by profiles defining the
boundaries of the area in question.
In practice, that means that upwards the
dune crest becomes the boundary (providing
there is no significant transport of sediment
across this line) while downwards various limiting standards will have to be considered. One
of these is "the limiting depth for active movement" (HALLERMEIER, 1972, 1981a, 1981b;
HANDS, 1979, 1980) which would come close to
2H bmax> where H bmax is the actual breaker
height of the highest waves within a certain
time period. The breaker height H, in relation
to breaker depth Db is Hb equal to 0.7-0.9 Db'
Up to that point the profile movement would
account for approximately 90% of the total profile movement, while the remainder may extend
to a "limited depth" of approximately 3.5 H bmax'
Various two-dimensional theories have been
proposed (EAGLESON, GLEEN & DRACUP
1961; HALLERMEIR 1972, 1981a, 1981b;
SWART 1974; TRASK 1955), but all under
idealized assumptions. The most practical way
of determining the limiting depth of profile
movement is by comparison between surveys,
like Figures 3 and 4. It is, however, a fact that
most profile-surveys have been seldom
Journal of Coastal Research, Vol. 4, No.4, 1988
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The Bruun Rule of Erosion
Table 2a.
, t
633
Profile characteristics.
a) Cross section, A
0-6m
6-9m
0-9m
1,500m2
2,500m 2
4,OOOm2
b) Width of the area, W
0-6m
6-9m
0-9m
350m
350m
700m
c) Mean depth, A/W
0-6m
6-9m
0-9m
4.0m
7.5m
5.5m
d) Steepness characteristic, A/W'
0-6m
6-9m
0-9m
10%0
5%0
8%0
I
2900m
1900
LEGEND
--71
------- 75
-5
77
~-~-79
--8\
-10
-15
-16m
"---
-20
m
Figure 3. Comparison of profile fluctuations on the Danish
North Sea Coast at Thyboroen. Data provided by the Danish
Coastal Directorate, Lemvig.
extended offshore far enough to determine the
limiting depth for profile movement (there are
a few exceptions). The problem of three-dimensionality still exists, where the third dimension
may be brought in by lateral, shore-parallel,
movement and a steep bottom, with gravity
influencing profile stability. The effect could be
either erosion or accretion on the lower part of
the profile. If it is erosion, the profile steepens
whereas with accretion it becomes flatter. The
difficulties involved in determining a practical
limit for the material exchange zone may best
be understood by considering some specific
examples.
MATERIAL BUDGET CALCULATIONSPROPER USE OF THE BRUUN RULE
Possible causes of shore erosion were previously discussed and this section now deals with
quantification of the sediment transfer accompanying a rise in sea level.
The
Bruun
Rule
(BRUUN,
1962;
SCHWARTZ, 1965, 1967) replaces more
involved theoretical or semi-theoretical models
by its simplicity. The basic assumption for the
model is that the profile, whatever its form,
maintains its shape during a period of sea level
rise. The rule has been tested accordingly by
various researchers (DUBOIS, 1976, 1980;
FISHER, 1980a, 1980b; ROSEN, 1978, 1980;
WEGGEL, 19790. If sea level rises "a" meters
and the width of the bottom influenced by the
sea level rise is "I" meters extending to depth
"h" meters, the shoreline recession s is determined by (Figure 1)
s·h=l·a
or
s
I· ex
h
(8) = (3)
The validity of the Bruun Rule has been discussed by many authors (ALLISON, 1980;
DUBOIS, 1976; FISHER, 1980b; ROSEN, 1978;
1980; SCHWARTZ, 1965, 1967; WEGGEL,
1979). BRUUN (1983, 1984) recommended
adjustments related to the grain size of the
shore material as well as to profile geometry
including a steepening of the outer part of the
profile as it is, for example, found off the southeast coast of Florida and at similar slopes,
ditches or trenches in many other parts of the
world.
To find the quantity of sediment eroded from
the profile to maintain its prior form, following
a rise in sea level, these steps are required:
(1) Survey of bottom profiles and comparison
between profiles as far offshore as in possible
considering variances (Table 2).
(2) Extreme wave analyses by which the ultimate or closure depth for exchange of material
between land and sea (Bruun-rule) is determined by 3.5 H b m a x (50-100) years (WES
research by Hallermeier and Hands).
Journal of Coastal Research, Vol. 4, No.4, 1988
I
Table 2b.
I
Bruun
634
Calculation of various standard deviations in investigations of changes in the beach profile.
I
Standard deviations (absolute and in %)
Area
J
area
2
prf
4
prf
16
prf
500m
area
2
prf
4
prf
16
prf
500 m
area
2
prf
4
prf
16
prf
190
13
130
9
100
7
50
3
250
10
180
7
130
5
60
2
130
3
90
2
70
2
35
1
25
7
20
6
15
4
40
30
9
20
6
10
3
35
5
25
4
20
3
10
1.5
500 m
Number of profiles
Cross Section
m2
%
Width Bottom (area)
m
%
Mean Depth
m
%
Steepness
Characteristic-%o
%
5
1
11
0.3
7
0.2
5
0.2
5
0.1
3
0.2
3
0.15
2
0.1
1
<0.1
<1
0.1
2
0.1
2
<0.1
1
<0.1
<1
0.6
6
0.4
4
0.3
3
0.15
2
0.9
15
0.6
12
0.5
10
0.2
4
0.4
5
0.3
4
0.2
3
0.1
1
(3) Analyses of bottom material revealing the
bottom widths with grain sizes and granulometric characteristics similar to the finest fraction (10%) of the material found in the beach
and dune material (heavy minearls excepted).
While 1 and 2 will provide a more distinct
depth, 3 may give a wide range. Having determined from the above a "closure depth", h
located at distance 1 from the shore, the quantity eroded from the profile to maintain its equilibrium shape for a sea level rise of "a" and a
shoreline recession "x" is then determined from
eqs. (3) = (8).
Thyboron Barriers, North Sea Coast,
Denmark (Bruun, 1954, a.b),
Take as an example Thyboron in Denmark
(Figure 3) where I up to 16 m depth is about
1500 meters. Erosion contributing to an apparent or expected sea level rise of 0.003 m/year
will amount to about 5 m 3/m of shore. The
remaining part of the erosion, which is approximately 50 m 3/m/year, is caused by a combination of waves and currents. The shoreline recession is computed as:
x =
45
0-9 m area
6-9 m area
0-6 m area
= about 2m/year.
The actual figure is somewhat less (about 1.5 m/
year) due to the steepening effect which groins
built on the shore have on the nearshore profile.
As such the Thyboron Barriers were analyzed
in great detail by BRUUN (1954, a.b) using an
overall 2-dimensional approach in this 3-
dimensional case as follows. On this approximately 20 krn-Iong shore, surveys in profiles
spaced about 600 m apart have been conducted
for more than 100 years. At first, profiles were
only taken to 6 m depth but later extended to
10 m and finally (since 1938) to 20 m depth (Figure 3). The movements in these profiles, including quantities eroded, are described in great
detail by BRUUN (1954 a and b). Due to slight
variations in survey accuracy and the fact that
a particular survey line represents a bottom
area of certain width, one has to accept some
variability in depths, profile areas, steepness
and, finally, in the calculated quantities based
on the movement in the profiles. Because of the
limited survey data available outside the 9 m
depth, Bruun was only able to compute variations up to 9 m depth for profiles spaced about
600 meters apart and extending 600 to 900 m
from the shore.
Table 2a gives average profile characteristics
for the characteristic parameters: Cross Section
up to a certain depth, corresponding width of
profile, its mean depth and its steepness =
depth/width. Table 2b shows standard deviations corresponding to Table 2a. Note that all
standard deviations decrease with increasing
profile dimensions and increasing number of
profiles. In this particular case it was found
that the limiting depth for onshore-offshore
movement of any importance was at 16 m, corresponding to about 2H b m a x for very unusual
storms of low frequency when waves of maximum height 8 m are not far from the breaking
depth, Db, at 16 m. From table 2b, one may con-
Journal of Coastal Research, Vol. 4, No.4, 1988
\
)
The Bruun Rule of Erosion
elude that standard deviations for depths up to
16 m would be < 0.1 m, or of the order of ± 0.05
m. The corresponding standard deviations for
bottom width, profile sectional area, and steepness are also given in table 2b, based on 0-6 m,
0-9 m and 6-9 m bottom areas. It may be
observed that all standard deviations are relatively small when there are 16 profiles.
The practical consequence of the above is that
one should compare profiles surveyed over various periods up to the depth where the average
difference in depths in the two profiles is less
than y'2x 0.05 m, i.e. approximately 0.07 m, or
at least less than 0.1 m. This, of course,
assumes a firm (sand) bottom. This definition of
"limiting depth" is practical but, as mentioned
above, it requires knowledge concerning the
development of depths out to a distance from
the shore of at least 2H b m a x where H b m a x designates extreme events of low frequency of occurrence, e.g. once every 50 years. Surveys, therefore must be undertaken when the sea is calm,
or profile records must be "smoothed" properly
by experienced surveyors and technique. Figure 3 compares profile fluctuations on the coast
of Thyboroen for a 10 year period, 1971-1981.
It may be noted that 10 m seems to be the limit
for "active movement". Scaling up to 100 years,
the figure may be 16-18 m while the "ultimate
depth" could be as high as 25-28 m. If profile
data like Figure 3 are not available, which
unfortunately is the normal case, one may try
to transfer experience from elsewhere, e.g. by
analysis of extreme wave events using a WEIBULL distribution (BRUUN, 1981; HOUMB,
1981). This would require multiplying the 50
year maximum wave height, which is about
1.7-1.8 H', by two, arriving at H' 50 max times
3.5, and then using that depth as the outer or
ultimate limit for exchange of material in the
active profile. 3-D effects, however, occur in the
deeper waters.
The Lake Michigan Coast
Considering another practical case, Figure 4
by HANDS (1980) shows an envelope of profiles
surveyed over a 9-year period (1967-1975) of
rising lake level, followed by stable water levels, on Lake Michigan. It may be observed that
profile changes were considerable up to about
11 m depth.
The Lake Michigan profile change occurred
635
during a period of 9 years of rapid rise of lake
level. There will be a phase difference in time
between water-level rise and profile adjustment, depending upon wave conditions during
storms occurring in Lake Michigan (HANDS,
1979, 1980). Considerably more information on
profile behavior in Lake Michigan is available
from the Waterways Experiment Station,
CERC, Vicksburg, Mississippi.
Regarding transversal movement due to sea
level rise, HANDS (1979, 1980) discusses the
"closure depth" in the profile in relation to the
Bruun Rule with special reference to the Great
Lakes, which during the period 1967 -1975
experienced a rapid rise in lake level. As
expressed by Hands, the theoretical depth up to
which bottom motion extends depends upon
wave height, wave steepness (period) and grain
size. Considering a certain number of years, the
highest wave during that period would be the
determining factor. This obviously means that
the closure depth for Hmax> 5 years is shallower
than the closure depth for Hmax> 50 years.
Relation of Bottom Adjustments to LongTerm Rise in Sea Level
In principle there is no difference between
short-term (Lake Michigan) and long-term rise
(the eustatic sea level rise). The latter, however, is very slow and consequently difficult to
trace directly. One may say that it is a natural,
integrated consequence of short-term movements, extended over a very long period. The
question is: "How long a period of time is
required to enable us to measure the reaction of
the profile to a long-term rise of sea level?"
Assuming a world-wide sea level rise of 5 mm
(0.005 m) a year, which is of the order of what
we may expect during the next decades, in 10
years this would attain 5 cm (0.05 m) or a figure
of the same magnitude as the standard deviaton
associated with our surveys. Most likely the
effect of such a rise on the profile development
would be so small that it could not be detected,
providing that none of the other aforementioned causes of erosion occurred during the
period under consideration. After 40 years the
effect could be seen clearly because a 0.2 m
change in sea level would be detectable within
the accuracy of the surveys.
The next question is: "Is it possible to obtain
advanced knowledge before the 40 years have
elapsed (assuming that no reliable survey data
Journal of Coastal Research, Vol. 4, No.4, 1988
,
636
,
1
Bruun
I
l
1
178
177'
17~
-~~ 175
,
I
)
~,
;;/174
w
j:;j
184
173
I
/
182
180
172
/
17~~0'
"
,
o
100
178
~ 176
~
~
j
w
~.n
,
"
200
"
300
-';;0
DISTANCE (M)
::~
174
172
170
168
166
164
162-100 0
t, , . , ,
200
' ,.
400
""""
600
eoo
I
•
•
I
1000
I
I
I
1200
•
I
•
,
1400
,~
1600
OISTANCE (M)
Figure 4.
Profile adjustments in Lake Michigan over a 9-year period (Hands, 1979).
from earlier surveys are available), or will we
be able to detect a slow offshore movement of
material in the profile?" One way in which this
could possibly be done would be by employing a
long-term fluorescent tracer. In such a study
tracers of various colors are placed at different
depths, which should all be outside the depth of
2H b m a x 5 years. If H b m a x 5 years is 5 meters, then
fluorescent tracers should be placed at 10 m, 12
m , 14 m , 16 m and 20 meters depth and the
movements of the tracers should be observed.
Most likely such tests would demonstrate considerable diffusion, as was experienced at the
EKOFISK tests in the North Sea at 72 m depth
(BRATTLELAND & BRUUN, 1975), so the
results may not be very reliable. A more direct
method would be to establish a grid system by
means of calibrated pegs placed in the bottom
and have diver observation of sediment surface
level over a large area. This would also indicate
the existence of longshore current-generated
bottom undulations, thereby material drift.
THE EFFECT OF THREE-DIMENSIONS
ON THE APPLICATION AND VALIDITY
OF THE BRUUN RULE
The Bruun Rule was proposed as a twodimensional model, but it is always used in
three-dimensions. What does this actually
mean? The fact is that true "academic twodimensionality" does not exist in nature,
because wave action is never exactly perpendicular to the shoreline. Even if it was, threedimensional phenomena would arise due to
wave breaking and the accompanying longshore currents parallel to shore. Two dimensions only exists (in its true sense) in a narrow
wave tank like the one Schwartz used for his
experiments in 1965. The fact that everywhere
in nature we find bottom profiles following the
equation y312 = P . x proves that the perpendicular to shore forces are by far the most important for profile development and geometries,
including bars and troughs. They do, of course,
not follow a simple equation.
With respect to the longshore drift, we use in
littoral drift technology the terminology "nodal
point" to describe the point or area of limited
length longshore where the resultant drift "left
minus right" is equal to zero. This is explored
in great detail in BRUUN (l954b, c; 1973,
BRUUN and SCHWARTZ, 1985). Computing
and adding the drift quantities numerically in
either direction one may wind up with a considerable quantity. In true three dimensionality
there is a resultant drift in one direction. Conside\-ing two cross sections a-a and bob located
Journal of Coastal Research, Vol. 4, No.4, 1988
The Bruun Rule of Erosion
a distance alb apart. The profiles within this
length of shore are, due to a slight shore line
curvature, subjected to wave action under a
slightly deviating angle of approach in breaking causing a drift in one direction at the predominant storms. The question arises: "Does
this influence the profile geometry?" It sounds
logical that it will do so, but in practice it is not
so. Profiles are, as mentioned above, always 3dimensional. But the resultant or predominant
longshore drift is always (much) smaller than
the total numerical drift and even much
smaller than the transversal drift in the profile
(BRUUN, 1954, 1955, 1973, 1983 and 1985 with
SCHWARTZ). From field observations we know
how profiles develop along the shore depending
upon which direction the shoreline turns compared to the wave action (BRUUN, 1954a, b, c).
If the shoreline turns away from the wave direction it will usually develop an equilibrium configuration where profile steepness only changes
very little.
If the shoreline remains straight and wave
action does not change, profile steepness may
increase (or decrease) slightly until a point
where the shoreline turns up against the waves
and steepness usually decreases due to deposits
of materials (BRUUN, 1954 b, c). Comparing
the profiles we find in all cases of uniform
material that the geometrical shape follows the
equation y3/2 = P . x. A flattening of the profile
in its deeper sections may occur, where the
shoreline turns up against the waves. This puts
a brake on the drift. Here the profile may
approach the equation s" = p. x, as also developed by BRUUN (l954b). Changes come gradually in all cases. The question now arises:
"How does this affect the Bruun Rule?"
Obviously the only change is that a quantity of
material must either be added to or subtracted
from the quantity 1. a = s.h (h = limiting depth
as defined earlier as the end of a "fading-out"
section). The total change of quantity in the
profile is the Bruun Rule's 1 . a (1 times a) plus
or minus the change in littoral drift capacity
between sections a-a and b-b. This quantity
may increase or decrease, even to a negative
value, but the basic principles of the rule
remain the same as long as bottom materials
are grains from fine sands and up. If not, adjustments, as mentioned later, will have to be
made. We are still talking about shores uninterrupted by such three-dimension elements
637
which may cause severe discontinuities in drift
modes and patterns, as they e.g. occur by radical changes in bottom materials like outcropping of harder materials in reefs or in headlands, or where the shore is "punctured" by a
river or tidal inlet, which disrupts the continuity of the drift, positively or negatively, and
thereby disturbs the normal profile development, so that the Rule's basic principles of profile development are not valid any longer. But
even then it should be remembered that as long
as profile geometries remain the same the Rule
can still be used, accepting the adjustment in
quantities caused by the change in drift quantities.
There are yet other factors which may influence the profile development. Materials may be
blown in by winds or gained or lost by diffusive
processes as explained in the following. Such
quantities may, however, be quantified and
included in the balance equations. The main
difficulty lies in an exact or even approximate
definition of "the offshore limit" for the wave
induced nearshore/offshore interaction process.
No equilibrium balance situation needs to exist
in the deeper offshore area, where currents are
offshore-originated. Bottom sediments in movement are of clay and silt size, occasionally fine
sand where currents are strong enough to carry
the material. Current-generated ripple marks
have been found at very great depths, e.g., at
5000 ft (1500 m) on the Blake Plateau off the
Carolinas and the US Southeast Coast, including indications of scour due to currents over
shells. Depending upon the grain size, the
"base" may extend to shallower or deeper
water. This refers to open sea coasts, where sediments of silt and clay size may be carried long
distances in suspension before deposition. In
defining the area of exchange between nearshore and offshore drifts, one therefore has to
consider the grain sizes and materials of certain
characteristics available on the shore. The finest parts of this material, which may stay in
suspension for long periods, therefore have to
be included in the materials balance equations.
This refers to a certain depth beyond which
fines may still be transported to much deeper
water for deposition (BRUUN, 1980; HANDS,
1980). Equation (3), s = Ia/h , then has to be
adjusted. If the factor r = ratio in percentage
of eroded material smaller than 0.06 mm to the
Journal of Coastal Research, Vol. 4, No.4, 1988
638
Bruun
Figure 7 presents a narrow shelf limited
oceanward by a steep slope extending to deep
water. In this case it will be necessary to introduce a "loss-function" at the outer edge, which
may be determined by topographic surveys or
by tracing. Loss of sediment to canyons is a
known phenomenon, e.g. in California. If the
percentage of loss can be evaluated, one has to
add another loss percentage R to eq. 3 making
it:
total amount of material eroded, the adjusted
equation (3) = (8) becomes:
s = la (1
+ r/lOO)/h
(9)
A "dividing line" between inshore and offshore
bottom areas therefore does not exist in a strict
sense of limits. At the most one may be able to
define a "dividing area" of a certain width,
which could in turn be explored by comparing
onshore and offshore sedimentary characteristics, by investigating the bottom fluctuations or
by tracing the movements of the bottom sediments. Such tracing would then have to be continued over a sufficiently long period of time to
establish the boundary area with a reasonable
accuracy. For the establishment of a qualitative
balance criterion it is, however, not absolutely
necessary to go into the smallest details of the
transverse exchange of materials. The offshore
area of exchange may be set by long-term study
of the variations in depth occurring. One may
then still consider the losses of a certain very
limited quantity of fines-if available-in the
beach and the nearshore sediments. Another
problem could be waves of material travelling
on the offshore bottom due to current action
(BRUUN, 1954 a, b). Figures 5 to 8 describe
four different situations. Figure 5 shows a
closed basin, e.g. a lake, where it is possible to
account for all material depositions on the bottom, as erosion and river discharges are known.
A rise of lake level causing erosion will therefore-with a certain phase delay-be balanced
by a bottom deposit corresponding to the yield
of sediment by erosion to the lake. This is a
"Great Lakes phenomenon" in the United
States, well described by several reports by the
WES of the USACE in Vicksburg, Mississippi.
Figure 6 shows a wide shelf, where all erosion
material or other material discharges, e.g. by
rivers, will be deposited on the shelf and for this
reason is traceable by mineral composition and
grain sizes.
Figure 5.
s
1· a
=
h
r
R
(1 + 100) (1 + 100)
TABLE
WATER
TABLE BEFORE
AFTER
RISE
RISE
Closed basin reaction to rising sea level (Bruun, 1983).
Journal of Coastal Research, Vol. 4, No.4, 1988
1
I
J
]
I
I
(9)
It may, of course, be difficult to determine R,
unless material deposits on a slope.
Figure 8 is a common case on shores with an
offshore platform generated during a period
with a lower sea level. It shows a relatively
steep slope some distance from shore which
becomes an area of deposition for sediments
"creeping or washed out from shore". This does
not necessarily mean that some fines may not
escape beyond that limit. On a sand shore this
will usually be a matter of minor quantities
only. The slope, in other words, indicates the
limit of the exchange area, but deposits on the
slope itself could be of considerable magnitude,
eventually causing slides as experienced e.g. off
Newfoundland.
Usually it will be possible to evaluate the
outer limit of the exchange area by more than
one method, e.g. using depth topography and
results of sedimentological investigation, and
thereby arrive at a reasonable result that is
useful for a practical quantitative evaluation of
erosion and deposition.
At the Hadera offshore terminal in Israel
(BRUUN, 1989) profile analyses using the statistical methods described in BRUUN (1954 a,
b), radioactive tracing and steel pegs placed in
grid system, have been used to determine the
offshore longshore drift versus the transverse
drift. On relatively exposed shores there are
WATER
l
I
I
I
I
I
J
1
J
)
J
I
I
I
The Bruun Rule of Erosion
Figure 6.
639
Wide shelf reaction to rising sea level (Bruun, 1983).
AFTER RISE
LOSS TO
Figure 7.
DEEP
WATER
Narrow shelf reaction to rising sea level (Bruun, 1983).
AFTER RISE
~
-<::t::J:::l'!
Figure 8.
Table 3.
1973).
-----------' - - BEFORE
tit'
!
I I
,
,
I
I
I!
RISE
t~DEPOSIT
SLOPE
Profile with deposit slope under rising sea level (Bruun, 1983).
Means of vertical, short-term fluctuations (Bruun,
Place
0-20 ft
(0-6 m)
20-30 ft
(6-9m)
Tokai, Japan
0.0 ft
(O.Om)
0.8 ft
(0.25 m)
1.4 ft
(0.40 m)
0.5 ft
(0.15 m)
1.5 ft
(0.45 m)
2.4 ft
(0.7 m)
Table 4. Test on bottom fluctuations at La Jolla, California
(Inman, 1956).
Depth in ft (m)
Mission Bay, California
Danish North Sea coast at Bovbjerg
many indications that the offshore limit may be
in the 50 to 70 ft (15-21 m) deep area on a longterm basis (TRASK, 1955).
How deep the limit is largely depends upon
the wave exposure. By splitting up bottom
areas in depth intervals, one many arrive at figures for variations in depth as shown in Table
3 (BRUUN, 1973) for three different areas
extending from relatively mild (Japan) through
medium (California) to exposed (Denmark,
North Sea) conditions. Recent Danish results as
mentioned earlier demonstrate that shore-term
70
52
30
18
Level
change in
ft (rn)
(21)
(17)
(9)
(5.5)
0.15
0.16
0.29
0.62
Sand level
occurrence in % of
obs. time
(0.04)
(0.05)
(0.10)
(0.20)
61
88
100
100
fluctuations on the North Sea Coast at Thyboron vanish at a depth of 16 m. This is about
2H m a x (BRUUN, 1973). Here 3-D effects also
occur.
Table 3, however, only demonstrates that seasonal fluctuations go deeper than 9 m. Field
examples from La Jolla Beach in southern California close to Mission Bay gave the results
listed in Table 4 (BRUUN, 1956; INMAN and
RUSNAK, 1956). It may be seen that fluctuations still take place at 21 m (70 ft) depth and
probably further out. This, however, may be a
result of offshore drift phenomena and could, of
course, also be survey variances. TRASK
Journal of Coastal Research, Vol. 4, No.4, 1988
I
640
Bruun
I
\
(1955), referring to the southern California
coast, maintains that the 30 ft (9 m) depth is the
boundary for the larger seasonal nearshore
fluctuations and that the 60 ft (18 m) depth is
the final boundary for the seasonal offshore
fluctuations of the bottom. The long-term
exchange zone referring to a "geological development" like sea level rises undoubtedly
extends deeper, beyond 20 meters.
It may be noted that fluctuations are small
from 17 m oceanward and that sand levels were
not always present at depths of 70 ft (20 m) and
52 ft (17 m). Even if the exchange area may
extend further out, the importance of the depositions beyond 21 m may be relatively small
and not disturb the "rule" to any practical
extent. If in certain areas material is carried
toward land from deeper waters by currentwave interactions this material must, of course,
be included in balance equations. But this is a
rare case. Normally bottom material decreases
in size oceanward. If the offshore bottom was a
source for natural nourishment of the nearshore area grain size would increase oceanward
from a certain point. It should then be possible
to trace the drift shoreward by grain-size distributions. This could happen in front of shores
which have been glaciated, such as Denmark.
The author, however, does not, at this moment,
know of even one case, where this happens (or
has happened), but he has experienced "negative cases" where it could possibly take place
but did not. An example is the west coast of Jutland, Denmark, where sea level for a very long
period, has stayed relatively stable compared to
land due to a glacial rebound which is now
being overpowered by sea level rise. Here
coarse glacial sand is available in the near-offshore. A similar result may be arrived at by a
different logic. If such a source existed it would
eventually run dry. Furthermore, an equilibrium slope would develop resulting in an equilibrium profile and this is one of the BruunRule's main assumptions. Adjustments of grain
sizes are from land and out. Finally, and most
logical for every representative of the physical
sciences, a rising sea level does not "generate
sand". It rather makes it worse for the waves to
pick up sand offshore even during the "rare
events". To consider the offshore bottom a
source of material for the nearshore, therefore,
is illogical, unless conditions like special current-wave interactions might make it possible.
If sea level drops, or it stays stable for a longer
time period, the offshore bottom may, until a
certain limit, offer material for the construction
of beach ridges, as pointed out by TANNER
(1988) in an article published in the Journal of
Coastal Research (4(1), 83-91). Ridges, however, may be built up any time-also during a
rising sea level on shores, where the littoral
transport slows down. This is very apparent,
where the beach drift material is coarse.
In his article on "Additional Sediment Input
to the Nearshore Region" published by Shore
and Beach (Oct. 1987), R. Dean points at the
nearshore bottom as a source of "considerable
likeliness" for beach nourishment even under a
rising sea level. Particular reference is made to
Florida. In doing so he makes several errors in
his references to the Bruun-Rule. Dean does not
seem to be acquainted with the rule's adamant
assumptions of "equilibrium profile" and "ultimate depth for exchange of material between
shore and offshore" (BRUUN, 1962, 1983). His
assumption of a slowly rising sea level during
the last 6,000 years is in opposition to facts. Sea
level has fluctuated. See e.g. the earlier cited
article by TANNER (1988). Dean refers to
ridges built during a rising sea level, not a lowering. They were, however, mainly built during
a lowering or a stable sea level interrupted by
rising levels. BRUUN (1962) explains that it
may be a considerable phase-lag between sea
level rises and profile reactions. As mentioned
earlier ridges may also be built any time as a
result of decreases of littoral drift capacities, as
we see it in spits, recurved spits, angular forelands, tombolos et cet. (BRUUN, 1954). It is also
very possible that material inside "the ultimate
depth" which is always located quit a distance
from shore, e.g. more than 20 m depth on the
Florida east coast and more than that on the
heavily exposed Danish North Sea Coast
(BRUUN,
1954,
1962;
BRUUN
and
SCHWARTZ, 1985) may be moved towards
shore if it happens to be surplus material, e.g.
deposited by currents on "the top of the equilibrium profile". Net-movements towards the
shore, however, stop when an equilibrium slope
or condition has developed. As mentioned by
BRUUN (1962) there might be a phase lag in
time between water table movements and the
reaction of the profile, particularly for gentle
slopes.
Dean's postulate on movement of material
Journal of Coastal Research, Vol. 4, No.4, 1988
I
)
I
The Bruun Rule of Erosion
towards shore in a laminar sublayer refers to a
horizontal bottom. If the bottom has a slope or
if it (as is usually the case for deeper waters offshore) is composed of fine particles which easily
become "fluidized" by high pressure gradients
caused by wave action, the movement may be in
an offshore direction in a kind of "density current". The experimental results by MURRAY
(1966) mentioned in connection with Figure 2
clearly demonstrated the tendency for finer
grains to move seaward, undoubtedly a result of
turbulent diffusion. ZENKOVITCH (1962) in
his field experiments in the Black Sea describes
how "the distribution of colored sand particles
showed the presence of a powerful bottom outflow seawards". A wind blowing with the waves
which is the normal case produces a current
with the wind at the surface and a return current at the bottom, further increasing the tendency for smaller grain sizes to move seaward.
BRATTELAND and BRUUN (1975) undertook
tracer experiments at 72 meters depth on the
bottom of the North Sea. They found that
material mainly moved up against the wave
action. Depth was about 3.5 times wave heights
during severe storms confirming Hands and
Hallermeier's earlier results. Inside the ultimate depth for any movement, as assumed by
the Bruun-Rule, material may move shoreward
by bottom creep (LONGUET-HIGGINS, 1953;
CARTER, LIU and MEl, 1973), but only until
the bottom has developed an equilibrium slope,
considering the effects of nearshore circulation
systems. Dean's statements on grain sizes are
also peculiar considering the pre-condition of
equilibrium profile until the ultimate depth in
the Bruun-Rule. Material does not start moving
shoreward outside the ultimate depth just
because sea level rose. It will probably rest even
better!! Finally it is a fact that most shores of
the world (80%) erode. In Florida, inlets are
mainly responsible. Not so in most other parts
of the world where sea level rise (in most cases)
is the only plausible cause of erosion (see e.g.
EVERTS, 1985).
A most recent contribution is probably the
paper by P. NIELSEN (Coastal Engineering,
vol. 12, no. I, 1988, pp. 43-62) titled "Models of
Wave Transport." Nielsen shows how waveinduced transport by non-breaking waves over
a horizontal, rippled bed can be presented by
three different models which are evaluated
through comparison with wave flume data. The
641
conclusion from this testing is "the simpler the
better." It is shown that the process in question
can be modelled without quantitative consideration of suspended sediment distribution.
Only a reference average concentration at the
bed is needed. On the other hand, classical diffusion models severely under-predict the transport of coarse sand in suspension because the
process is much more organized than diffusion.
In the conclusion of his paper Nielsen states
that the influence of wave shape, or the shape
of the oscillating water velocity movement, on
the direction of material transport is determined by the relative maximum and minimum
velocities and by entrainment coefficients. This
refers to fine sands but for coarser sands acceleration effects need to be included.
Nielsen's figures demonstrate how sands <
0.2 mm had a predominant transport up against
the direction of wave propagation that means
opposite to the largest absolute velocities. This
is a confirmation of the field results by MURRA Y (1966). "Fine sand" is interpreted relatively as d/A < 0.004, d = grain diameter. A =
semi excursion fundamental mode. The finest
non-cohesive particles move until they come to
an "ultimate rest." Before then the relatively
coarser grains have stopped movements. In this
respect it is interesting to note the classical
results by SHIELDS (1936), BROWN-KALINSKE (1950) and BROWN-EINSTEIN (1950).
According to Shields bed load transport is proportional to liD where D is grain diameter,
BROWN -KALINSKE got the same and
BROWN-EINSTEIN lID 2 / 3 • Nielsen's results
may be influenced by a shallow water condition
of rather steep ripple marks. For more rounded
ripples, which may be found in deeper waters,
the predominance of offshore transport
decreases and may turn to onshore for a horizontal bottom. When the bottom starts sloping
updrift is again reversed, particularly if wind
generated return flows of bottom waters enter
the picture. In deeper waters like "the ultimate
depth" three-dimensional phenomena like current-generated moving sand waves or undulations may have arrived on the scene. They were
already noted by divers on the Danish North
Sea Coast off Thyboron in the 1940's and were
reported to move parallel to shore. Recent
research seems to prove that they may also
move under an angle probably with a meandering current. If they move in at one place due to
Journal of Coastal Research. Vol. 4. No.4. 1988
642
Bruun
such current they must move out again somehow at a different place simply due to the equation of continuity for flow. The magnitude of
transport however is small.
Table 5.
1971.
CONSOLIDATION
Total
Eustatic
mm/year
Various geological settling or tilting theories
have been proposed. Tiltings are usually associated with fall areas, like the Californian
Pacific which seems to tilt up causing less relative sea level rise, 1-2 mm/year compared to
the Atlantic (2-4 mm/year). The same is true
for most of the Pacific coast where the San
Andreas fault line is close to shore. Tectonic
movements may also be of volcanic nature, such
as part of the Icelandic, Italian, and Japanese
shores. Most Scandinavian shores are still rising relative to sea level as a result of glacial
rebound. This includes the northernmost part of
Jutland, but not Skagen, the northernmost
point where, as explained by Hauerbach in a
1988 article published in the Journal of Coastal
Research (4(4)), subsidence due to compaction of
deep water silts in a finger of the Norwegian
Trench apparently takes place causing a negative balance of 3-5 mm/year. This assumes no
glacial rebound at Skagen. Most of Sweden,
apart from the southernmost province of
Skaane, is rising. This is very evident in the
west coast province of Bohualan south of the
town of Gothenborg where Svante Arhenius'
famous experiments on the relative movement
of land and sea were begun in 1890 with the
drilling of holes in rocks facing the sea, and in
parts of the Oslo Fiord in Norway. In other sections of the Scandinavian peninsula, such as
the Norwegian Atlantic coast, movements have
largely stopped, but it is undoubtedly continuing in the northernmost part of the Fenno-Scandinavian peninsula. The character of the bed
rock may be responsible for the differences in
recorded movements.
In the Mediterranean area the relative movements may have been influenced by subsidence,
e.g., at Venice, where the withdrawal of water
and gas from underground has undoubtedly
been responsible for the recorded sinking. Table
5 (BRUUN, 1983) shows that movements at
Venice have not been solely eustatic. On the
other hand, Table 6 indicates that the Venician
subsidence now has stopped and a small recovery may take place following reduced with-
Rises of sec levels during the period about 1930 to
Venice
Trieste
0.8 ft (25 em)
0.3 ft (10 em)
6mm
0.3 ft (10 em)
0.3 ft (10 em)
2.5mm
Florida East
Coast
0.4 ft (12 em)
0.4 ft (12 em)
3.0mm
drawal of groundwater. This tendency has continued.
Other areas of subsidence include parts of
Holland, where filling on top of silt and clay
layers has been undertaken, the Hokkaido
Island in Japan where sinking is mainly due to
recovery of gas from the subsurface, and part of
the Los Angeles area in California where
extraction of oil has taken place. Gas extraction
from the sea bottom is expected to cause considerable subsidence of the Dutch North Sea coast
in local areas (Wiersma, pers. communication,
1988).
DO LOSSES OR GAINS OF MATERIALS
NOT RELATED TO SEA LEVEL RISE,
TECTONIC MOVEMENTS, AND
SUBSIDENCE INFLUENCE COASTAL
GEOMORPHOLOGY AND THE
APPLICATION OF THE BRUUN RULE
TO EXPLAIN SHORE EROSION?
The answers to the above questions may be
summarized briefly as follows.
(1) In the case of tectonic movements its
known value is added to or subtracted from the
sea level movement known from adjoining
shores not subjected to tectonic movements.
(2) Subsidence, if known, is added to the
known sea level rise. Some extraordinary surface sinkings, e.g. Hokkaido (Japan), Long
Beach (California), local areas in Holland, and
Table 6. Comparative annual movements at Venice, Trieste
and Florida in mm/year. Period 1930 to 1971.
Period
Venice
Trieste
Florida East
Coast
1930-1950
1942-1962
1961-1971
1971-1980
7.5
3
4.5
-1.6*
2.5
2.5
(2.5)
5.0
4.0
3.0
* Pumping of ground water prohibited, causing a
temporary uplift.
Journal of Coastal Research, Vol. 4, No.4, 1988
The Bruun Rule of Erosion
Venice (Italy) are in part caused by man's activities. In either case, (1) or (2), the shoreline
movement will adjust itself to the actual relative movements land/sea. Consequently, uplifts
will cause a relatively protruding shoreline in
the local area of uplift and an indented shoreline where settling or downwarping occurs.
These phenomena are well known, e.g. from
California, the Danish North Sea Coast, Italian
shores on the Adriatic and from Scandinavian
shores where relative but uneven uplifts take
place. Protruding areas will have a groin-effect
on adjacent shorelines. Settling or sinking
areas will give a trap effect, in either case
necessitating adjustments in the use of Bruun
Rule based on known movements.
643
overall stability of profile geometries are maintained. Settling or consolidation of shores on
softer materials, as well as erosion of softer bottom materials of silts and clays which when
eroded diffuse away to deeper bottom areas will
influence the rate of development, but not profile geometry if the bulk part of the material is
sand. Such areas will act as material traps
influencing the rate of erosion of not only the
area itself (like many barrier coasts in Denmark and Florida) but the adjoining shore as
well. The Rule is still applicable when adjusted
to the actual relative movements land/sea level.
Profile geometries do not change. The perpendicular-forces are the overwhelming ones!
GENERAL CONCLUSION ON THREEDIMENSIONAL EFFECTS
IS THE RULE APPLICABLE UNDER
CONDITIONS OF A FALLING SEA
LEVEL?
The Bruun Rule (1962) was proposed as a twodimensional model for profiles which are subjected to wave action of perpendicular incidence
or for the "so-called nodal areas" of littoral
drift. In practice this a statistical not physical
quantity. Moving away from these nodal areas
bottom profiles retain their geometrical shape
and steepness, if the shoreline, as at the Danish
Thyboroen Barriers (BRUUN, 1954 b,c), turns
slightly with the waves so that total drift
increases with the gradual increase of the angle
of incidence of waves or the breaker angle. If
the shoreline is straight or turns slightly up
against the wave action, profile steepness tends
to decrease gradually as seen from the Danish
North Sea Coast and from Lake Michigan. This,
however, only changes the p-value in the
e.q.: y3/2 = P . x. As proven by numerous field
surveys (United States, Denmark, Holland) the
bottom profiles react to wave action causing
nearshore/offshore exchange of materials up to
depths of 3-4 H'', where H b refers to waves
occurring at intervals once every 20-50 or up
to 100-200 years. Tectonic movements will of
course interfere with the rate of profile movements relative to sea level change, but only if
the shore is built up of alluvial materials, not
with the profile geometry. The Bruun Rule still
is applicable when used in relation with the relative land/sea level movements. Littoral drift
barriers such as headlands and tidal inlets may
cause major deviations in the development of
bottom profiles. The Rule is only applicable if
Falling of sea level has taken place in various
geological periods, when interglacial ice melts
were replaced by glacial periods. Tectonic
uplifts have also occurred and this, of course,
caused adjustments of beach and bottom profiles to the new situation. At a certain depth
with a slope a, adjustment to a steeper slope>
a may then result. With reference to Figure 9,
a preliminary inspection seems to support the
view that the profile development now is going
to be the opposite of the development shown in
Figure 1 for a sea-level rise. But second
thoughts cause some revision. Firstly, the eroding and accreting forces are very different. In
the nearshore area a falling sea level will generally cause erosion of the bottom and accretion
on the shore. This is the universal geological
and geomorphological experience. On the
beach, ridges may then build up (one after the
other) as sea level continues to fall. Beach
ridges are parallel to the shore and well known
from the huge beach-ridge systems found on
shores all over the world, particularly in coarse
materials. This, however, mainly refers to steep
shores. In the case of more gently sloping profiles, offshore bars may not only move seaward,
but new bars or shoals may appear. The material in all cases comes from the surrounding
shores and bottoms. It may come from the longshore drift building up marine forelands
(BRUUN, 1954b; ZENKOVICH, 1967) or from
offshore bottoms (TANNER, 1988). This
requires adjustments of the general Bruun-
Journal of Coastal Research. Vol. 4. No.4, 1988
644
Bruun
SEA LEVEL
BEFORE
LOWERING
- - - - +--.y\..--~o:__---------------~------. .,,~
"
SEA LEVEL
AFTER
LOWERING
ia
t
LOWERING
PROFILE
Figure 9.
AFTER
LOWERING-~~
---- I
Profile development following a lowering of the sea level (Bruun, 1983).
Rule and its two-dimensional assumptions, as
explained above. Maintaining profile equilibrium geometry the Rule is still valid even for
an accreting shore under a rising sea level, with
proper "amendments".
LATEST DEVELOPMENTS IN
RESEARCH
An extremely informative bibliography of
sea-level changes along the Atlantic and Gulf
coasts of North America has recently been published by LISLE (1982). The comprehensive
material reported (200 references!) leaves no
doubt that sea level has fluctuated dramatically
in known geological history and that it has had
an equally dramatic influence on the distribution of land and water masses and surfaces.
There is little reason to assume that these
events are not a continuing process caused by
differences in energy emissions by the sun and
the accompanying reactions on temperatures on
the earth, possibly including some changes in
the air chemistry. One may get an impression
of the comprehensive work by LISLE (sponsored
by the Office of Naval Research) by the following lines in direct quotation:
"Sea level changes are also accompanied by
morphologic reactions on the bordering
coastlines. The evolution and migration of
barrier islands, capes, and other coastal
features appear to be directly related to sea
level changes; thus they have been included
in this bibliography. The Bruun Rule, proposed by Bruun, 1962, demonstrates how a
shoreline is in equilibrium with its nearshore bottom. A rising sea level can cause
that equilibrium to change, forcing immediate readjustments. Hoyt, Otvos, and
Kraft et al., among others, have examined
the origin and migration of barrier islands.
Estuaries and salt marshes have been considered against the light of sea level rises
by Keene, Redfield, Kraft and Caulk, SWift,
Kayan and Kraft, Froomer, Rampino and
Sanders, and others."
The "main lines" of relatively slow geological
development, therefore, leave no doubt. With
respect to the short-term development, a great
number of variables exist, as is explained earlier in this paper. If the shore is composed of
uniform material of sand size from the dunes to
depths of say 20 to 30 m and it is located in a
neutral area with respect to littoral drift, conditions are optimum for fulfillment of the rule.
If not, complications may arise. DUBOIS (1982)
considers "Relation among Wave Conditions,
Sediment Texture, and rising Sea Level". His
remarks concentrate on the influence of sediment characteristics. If an eroding profile runs
into very coarse material or if, conversely, it
proceeds into very fine (silt and clay) materials,
the profile has to adjust itself to a new situation
and it may be difficult to check on the validity
of the rule unless all the new characteristics are
considered. Some authors (e.g. HALLERMEIER, 1981b) have already tried to do this in
simple cases. Short-term evidence must of
necessity rely upon conditions of an ideal
nature like those reported from the Great Lakes
(HANDS, 1980). We are, in other words, forced
to develop an "envelope of possibilities" with
factors carrying various weights, but comprehensive research on profile developments giv-
Journal of Coastal Research, Vol. 4, No.4, 1988
645
The Bruun Rule of Erosion
ing physical reasons (examples: BRUUN, 1954
a, b; WEGGEL, 1979; HANDS, 1980; HALLERMEIER, 1981b) and mathematical "commonsense reasonings" (ALLISON, 1980, 1981) all
point in the same direction which is: Nature
tries to establish a new equilibrium condition
by erosion of the beach and nearshore bottom
and deposition offshore of the material eroded.
Some, perhaps all of it, will stay on the nearshore, predominantly wave-generated bottom.
Other materials may move farther away. With
the experience already gained in past geological history, combined with recent experience on
sea-level rises which are evident and considering the unavoidable fact that this will cause a
long-term, long-lasting erosion, American geologists have expressed concern in their statement of March 1981 entitled "Saving the American Beach: a Position Paper by concerned
coastal Geologists", now widely distributed in
the United States. Their concern is a product of
much thought about the consequences of a continued losing battle against beach erosion. We
shall, of course, accept this, but we do not necessarily need to defend ourselves or to win the
battle everywhere, if we can win it where it is
most needed. Coastal protection technology is
still advancing ("God created the Earth, but the
Dutch created Holland"). At some strategic
points we will win or at least maintain a condition of equilibrium. In other, much larger,
areas we must organize the retreat in an
orderly manner and with determination. So we
must zone the shore and just wait until sea level
retreats again, as it has done before innumerable times (BRUUN, 1983).
The 1986 Environmental Protection Agency
(EPA) conference in Washington, D.C., on
"Effects of Changes in Stratospheric Ozone and
Global Climate" contributed to the understanding of the large scale changes in the atmosphere
and their climatic effects including warming
trends and the resulting influence on the sea
level, thereby on coastal erosion. Recent reports
including the EPA report on the development of
erosion at Ocean City, Maryland, work by
EVERTS (1985), and report of the Louisiana
Wetland Protection Panel (April, 1987) all present objective quantified views on the development which is progressing of great concern to
all coastal scientists and engineers and to a
number of coastal communities.
SUMMARY AND CONCLUDING
REMARKS
The Bruun Rule proposed in 1962 seems to
have an overall general validity. But it is twodimensional and therefore care should be taken
in expanding it three-dimensionally. The twodimensional boundary conditions in relation to
the composition of beach and bottom materials
and to the bottom geometry extending to the
ultimate depth of exchange should be evaluated
and accounted for in the material balance
budget and equations. For this bottom fluctuation, statistics and tracing may be used. The
theory is firstly one of erosion, not accretion. It
is evident in a laboratory tank and under similar simple field conditions but in all other cases
it must be subjected to realistic adjustments of
its material balance assumptions and equations.
ACKNOWLEDGEMENT
Thanks are extended to Maurice L. Schwartz,
Dean, Washington State University (Bellingham), for beneficial collaboration throughout the years.
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n RESUMEN D
La regia de Bruun de erosion, denominada asi por geomofologos americanos (Schwartz, 1967) fue publicada por primera vez en
1962 (Bruun, 1962) y, en breve, se refiere al balance de sedimentos transversal de un perfil de playa a largo plazo. La regia se
basa en la hipotesis de la existencia de un balance de sedimentos entre: (1) playa y (2) perfil del fondo del mar (exterior a la playa).
La figura 1 es una representacion esquematica del efecto, una traslacion del perfil una distancia s despues de una elevacion de a
del nivel del mar, produce una erosion de la linea de costa y un deposito de sedimentos. Este tema ha sido tratado teoricamente
(Hallermeier, 1972; Allison, 1980; Bruun, 1980, 1983) y experimentalmente en la naturaleza (Bruun, 1954a,b, 1962, 1980, 1983,;
Dubois, 1976; Rosen, 1978, 1980; Weggel, 1979; Fisher, 1980; Hands, 1980; Schwartz, 1965, 1967, 1979). Con posterioridad la
Regia ha sido utilizada en diferentes informes sobre erosion de playas en Ocean City, Maryland, "Impacto potencial de la elevacion
del nivel del mar en la playa de Ocean City, MD" publicado por la EPA, Octubre 1985 y en un articulo de Everts (1985),
La regIa ha sido usada a veces inrlescriminadamente sin tener en cuenta sus limitaciones. En primer lugar debe tenerse en
cuenta que la regIa es basicamente bidmensional, pero se apl ica casi siempre con caracter tridimensional. Esto ha causado un
gran numero de malas interpretaciones. Usada objetivamente la RegIa ofrece una linea de referencia sobre todos los desarrollos
que ocurren en el perfil basico en relacion con el nivel del mar observado.
En este articulo se discuten condiciones de contorno, desviaciones y ajustes que hacen la RegIa uti l para interpretar los fenomenos observados de una manera cuantitativa.-Department of Water Sciences, University of Cantrabria, Santander, Spain
Journal of Coastal Research, Vol. 4, No.4, 1988